1. Field
Certain aspects of the present disclosure generally relate to a wireless communication and, more particularly, to fast denoising of a received signal.
2. Background
Ultra-Wideband (UWB) communications are radio communications that use a frequency bandwidth larger than 500 MHz. In comparison to narrow-band communications which rely on modulation of a carrier frequency, the large bandwidth of UWB communications allows sending signals with features well-localized in time. If a signal is more localized in time, then it is more spread in frequency. This allows communications based on pulses, while information can be encoded in a distance between pulses (i.e., a Pulse Position Modulation: PPM), in a pulse amplitude (i.e., a Pulse Amplitude Modulation: PAM) or in a pulse width (i.e., Pulse Width Modulation: PWM). One of the key advantages of pulse-based communication is ability to precisely localize time of arrival of the information (i.e., arrival of the pulse).
A signal at the UWB receiver is typically based on a pulse signal corrupted by stationary and non-stationary noise and by various channel effects, wherein the pulse signal can be different than a periodic sinc (i.e., Dirichlet kernel) shaped pulse signal. On the other hand, a parametric Finite Rate of Innovation (FRI) processing technique that can be applied after the UWB processing requires an input signal based on the periodic-sinc signal. Therefore, the received UWB pulse signal needs to be properly adjusted (i.e., equalized) before being processed by the FRI module. However, the equalized pulse signal at the input of the FRI module can still be corrupted by a prohibitively high level of noise.
The well-known Cadzow iterative algorithm can be used as an integral part of the FRI processing for denoising of the input signal of the FRI receiving module. The standard Cadzow algorithm provides, given a Toeplitz Hermitian square matrix of dimension N×N associated with the noisy signal, a Toeplitz Hermitian square matrix of the same dimension with rank K, where K<<N . In order to achieve this low-rank approximation (i.e., signal de-noising), the standard Cadzow algorithm performs eigenvalue decomposition (EVD) of the Toeplitz Hermitian square matrix of dimension N×N, and reconstructs the “best” rank K approximation by keeping only the K principal eigenvalues and eigenvectors. The reconstructed rank-K matrix is made Toeplitz by averaging its diagonals. This process is iterated until convergence. However, the standard iterative Cadzow algorithm is slow and computationally complex.
A method is proposed in the present disclosure to speed up the standard Cadzow denoising algorithm and to lower its computational complexity.